Subsequence converging to inf of sup

Question

Let $(x_n)$ be a bounded sequence, and for each $n\in\mathbb{N}$ let $s_n=\sup\{x_k:k\geq n\}$ and $S=\inf\{s_n\}$. I need to show that there exists a subsequence of $(x_n)$ that converges to $S$.
I cannot proceed in the question as I am not sure what implication $s_n$ and $S$ have in terms of series convergence

Answer 1

Note that $s_n$ is monotonically decreasing to $S$ as $n \rightarrow \infty$. Since $s_{n}$ is the supremum of $\{x_k : k \geq n\}$, we can inductively pick $x_{n_i}$ with $n_{i+1} > n_{i}$ such that $x_{n_i} > S + 1/i$. It follows that $x_{n_i}$ goes to $S$ as $i \rightarrow \infty$.